Commutative unital Banach algebra with nilpotent elements What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?
 A: If you consider a closed (two-sided) ideal $I$ of the Banach algebra $A$, then you can endow $A/I$ with the quotient norm
$$
\|a+I\|=\inf_{x\in I}\|a+x\|
$$
and this norm makes $A/I$ into a Banach algebra (see proof on PlanetMath).
If you now consider $a\in A$ and the closure $I$ of the ideal generated by $a^n$, then $a+I$ will be nilpotent in $A/I$.
Another example. Let $A$ be a commutative Banach algebra and consider $B=A\oplus A$ endowed with the norm $\|(a,v)\|=\|a\|+\|v\|$. Then $B$ is a Banach space. Define an operation on $B$ by
$$(a,v)(b,w)=(ab,aw+bv)$$
It's easy to check that this is a good multiplication turning $B$ into an algebra. For the norm we have
\begin{align}
\|(a,v)(b,w)\|&=\|(ab,aw+bv)\|\\
&=\|ab\|+\|aw+bv\|\\
&\le\|ab\|+\|aw\|+\|bv\| &&\text{(triangle inequality)}\\
&\le\|a\|\,\|b\|+\|a\|\,\|w\|+\|b\|\,\|v\| &&\text{($A$ is a Banach algebra)}\\
&\le\|a\|\,\|b\|+\|a\|\,\|w\|+\|b\|\,\|v\|+\|v\|\,\|w\|\\
&=(\|a\|+\|v\|)(\|b\|+\|w\|)\\
&=\|(a,v)\|\,\|(b,w)\|
\end{align}
Thus $B$ is a Banach algebra and every element of the form $(0,v)$ is nilpotent, because $(0,v)^2=(0,0)$.
A: Upper triangular Toeplitz matrices give another example, i.e., the unital subalgebra of the $n$-by-$n$ matrices generated by the matrix with $1$s on the superdiagonal and $0$s elsewhere.
More generally, take any (nonzero) nilpotent element $a$ of a unital Banach algebra $B$, then take the unital Banach subalgebra of $B$ generated by $a$.  
